Productivity

In addition to conducting our analysis with biomass as the measurement of ecosystem function, in this document we report our results using net primary productivity (NPP).

Mirroring the manuscript’s central analysis, our models for the across-treatment effect were encoded as: Productivity ~ -1 + Stage + Stage:Shannon, and the within-treatment effect was encoded as: Productivity ~ -1 + Richness:Stage + Richness:Stage:Shannon. All models successfully converged, with Rhat values of 1.0, and posterior predictive checks (PPC) were used to visually validate the model fits.

1 Figure 2 - Counter-gradient

The relationship between Shannon diversity and productivity was qualitatively similar to that of Shannon diversity for the majority of the models (4/6).

In Forest2, while the within-treatment slopes are consistent between biomass and productivity, the across-treatment slopes differ. During the with stage, the across-treatment slope is insignificant with biomass, but significant and slightly positive for productivity. During the without seed inflow phase, the significant and negative across-treatment slope of the relationship between biomass and Shannon diversity becomes insignificant for productivity.

Dryland displays the most variation between the two measures of ecosystem functioning. The predominant difference is shows in the within-treatment slopes, as they flip from being significantly positive to significantly negative in both the with and without seed inflow phases. Secondly, while the across-treatment slope is insignificant during the without seed inflow phase for biomass as the ecosystem functioning, for productivity the slope is significant and positive. The reason for this change is clerical, because while seed biomass is incorporated into the productivity calculations, it is left absent from the total biomass calculations.

2 Figure 3 - Across-treatment effect

Considering the relationship between our measure of the internal coexistence processes within each model and the across-treatment effect of realized diversity on the focal ecosystem function (either productivity or biomass), we find that the aggregate patterns are nearly identical between ecosystem functions.

3 Figure 4 - Within-treatment effect

Considering the relationship between our measure of the internal coexistence processes within each model and the within-treatment effect of realized diversity on the focal ecosystem function (either productivity or biomass), we find that the aggregate patterns are nearly identical between ecosystem functions.

4 Model validation

This section of the document describes the statistical models’ validation, using Shannon diversity as the focal biodiversity metric and productivity as the focal ecosystem function.

Important terms:

4.1 Grass1

Clark, A. T., C. Lehman, and D. Tilman. 2018. Identifying mechanisms that structure ecological communities by snapping model parameters to empirically observed trade-offs. Ecology Letters 21:494–505.

4.1.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: productivity ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow                2.36      3.63    -4.71     9.64 1.00     2362     2294
## StageWithoutseedinflow           -12.23      3.95   -19.90    -4.60 1.00     1903     1847
## StageWithseedinflow:Shannon       17.76      1.52    14.76    20.78 1.00     2433     2253
## StageWithoutseedinflow:Shannon    25.70      1.93    21.93    29.44 1.00     1792     1873
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    23.18      0.59    22.08    24.40 1.00     2808     2294
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##    Estimate  Est.Error      Q2.5     Q97.5
## R2 0.296204 0.02350415 0.2492173 0.3402776

4.1.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

4.1.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.1.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

4.1.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.1.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: productivity ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedinflow                83.12     25.51    33.18   134.09 1.00     4919     2944
## Ninitial4:StageWithseedinflow               151.76     24.44   105.83   199.35 1.00     5105     2673
## Ninitial8:StageWithseedinflow               166.70     19.10   130.40   203.15 1.00     4500     2749
## Ninitial16:StageWithseedinflow              250.46     17.52   216.04   284.94 1.00     4987     2903
## Ninitial32:StageWithseedinflow              278.91     19.71   239.91   317.22 1.00     4908     2609
## Ninitial2:StageWithoutseedinflow             23.07     10.19     3.56    43.26 1.00     5483     3074
## Ninitial4:StageWithoutseedinflow             64.69     14.55    36.70    92.63 1.00     5670     2851
## Ninitial8:StageWithoutseedinflow             89.13     14.22    61.36   117.73 1.00     4971     2888
## Ninitial16:StageWithoutseedinflow           184.81     17.91   149.85   219.44 1.00     4928     3186
## Ninitial32:StageWithoutseedinflow           189.47     23.32   142.71   235.02 1.00     6235     2422
## Ninitial2:StageWithseedinflow:Shannon       -36.70     15.88   -68.43    -5.39 1.00     4912     2900
## Ninitial4:StageWithseedinflow:Shannon       -54.37     11.18   -76.18   -33.14 1.00     5154     2872
## Ninitial8:StageWithseedinflow:Shannon       -47.49      7.19   -61.33   -33.68 1.00     4293     2852
## Ninitial16:StageWithseedinflow:Shannon      -64.87      5.96   -76.64   -53.25 1.00     4984     2807
## Ninitial32:StageWithseedinflow:Shannon      -64.88      6.23   -76.98   -52.60 1.00     4930     2644
## Ninitial2:StageWithoutseedinflow:Shannon     -3.36      6.83   -16.61     9.79 1.00     5501     3010
## Ninitial4:StageWithoutseedinflow:Shannon    -19.36      7.47   -33.62    -5.04 1.00     5613     2798
## Ninitial8:StageWithoutseedinflow:Shannon    -23.93      6.33   -36.57   -11.55 1.00     4946     2917
## Ninitial16:StageWithoutseedinflow:Shannon   -52.76      7.38   -67.14   -38.46 1.00     4975     3208
## Ninitial32:StageWithoutseedinflow:Shannon   -44.28      8.90   -62.12   -26.38 1.00     6256     2552
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    15.04      0.44    14.21    15.95 1.00     7394     3056
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.6994955 0.01128394 0.6755486 0.7195096

4.1.2.1 Posterior predictive checks

4.1.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.1.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

4.1.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.2 Grass2

Turnbull, L. A., J. M. Levine, M. Loreau, and A. Hector. 2013. Coexistence, niches and biodiversity effects on ecosystem functioning. Ecology Letters 16:116–127.

4.2.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: productivity ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow               44.52      1.61    41.36    47.70 1.00     2125     2101
## StageWithoutseedinflow            37.52      1.79    33.95    41.11 1.00     2081     2146
## StageWithseedinflow:Shannon        8.71      0.56     7.57     9.80 1.00     2156     2175
## StageWithoutseedinflow:Shannon    15.08      0.79    13.49    16.64 1.00     1978     2150
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    12.22      0.31    11.60    12.84 1.00     2951     2255
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.4415411 0.01983217 0.4003317 0.4789923

4.2.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

4.2.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.2.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

4.2.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.2.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: productivity ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedinflow                84.84     33.26    20.52   149.85 1.00     5106     2557
## Ninitial4:StageWithseedinflow               135.66     35.43    67.21   206.42 1.00     4361     2258
## Ninitial8:StageWithseedinflow               133.36     31.45    73.46   194.25 1.00     4089     2679
## Ninitial16:StageWithseedinflow               94.00     49.50    -1.22   188.99 1.00     4620     2745
## Ninitial32:StageWithseedinflow               41.88     60.77   -78.97   159.87 1.00     4763     2773
## Ninitial2:StageWithoutseedinflow            -16.60     23.17   -60.09    29.90 1.00     4681     2505
## Ninitial4:StageWithoutseedinflow             27.33     13.80    -0.08    54.41 1.00     5324     2756
## Ninitial8:StageWithoutseedinflow             38.95     10.86    17.72    60.86 1.00     4866     2693
## Ninitial16:StageWithoutseedinflow            47.75     11.36    25.45    69.78 1.00     4839     2670
## Ninitial32:StageWithoutseedinflow            72.53     14.90    43.30   101.07 1.00     4600     2721
## Ninitial2:StageWithseedinflow:Shannon       -16.05     20.16   -55.34    22.78 1.00     5101     2574
## Ninitial4:StageWithseedinflow:Shannon       -29.65     15.35   -60.30     0.14 1.00     4384     2230
## Ninitial8:StageWithseedinflow:Shannon       -20.19     10.86   -41.24     0.50 1.00     4100     2854
## Ninitial16:StageWithseedinflow:Shannon       -5.12     13.97   -32.00    21.75 1.00     4601     2885
## Ninitial32:StageWithseedinflow:Shannon        8.38     14.48   -19.83    37.18 1.00     4825     2774
## Ninitial2:StageWithoutseedinflow:Shannon     46.15     14.33    17.11    73.36 1.00     4697     2508
## Ninitial4:StageWithoutseedinflow:Shannon     22.04      7.50     7.37    36.94 1.00     5408     2919
## Ninitial8:StageWithoutseedinflow:Shannon     17.58      4.91     7.72    27.16 1.00     4887     2792
## Ninitial16:StageWithoutseedinflow:Shannon    12.12      4.16     4.10    20.37 1.00     4820     2662
## Ninitial32:StageWithoutseedinflow:Shannon     3.27      4.45    -5.22    12.02 1.00     4557     2702
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     9.04      0.25     8.57     9.54 1.00     5731     2754
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.4943489 0.01956404 0.4532455 0.5312383

4.2.2.1 Posterior predictive checks

4.2.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.2.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

4.2.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.3 Grass3

May, F., V. Grimm, and F. Jeltsch. 2009. Reversed effects of grazing on plant diversity: The role of below-ground competition and size symmetry. Oikos 118:1830–1843.

4.3.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: productivity ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow               46.06      1.40    43.31    48.82 1.00     1952     2034
## StageWithoutseedinflow            46.05      1.47    43.25    49.02 1.00     2143     2182
## StageWithseedinflow:Shannon        2.70      0.52     1.68     3.70 1.00     1945     2316
## StageWithoutseedinflow:Shannon     2.87      0.59     1.65     3.98 1.00     2174     2135
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     9.96      0.26     9.47    10.49 1.00     3156     2237
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##      Estimate  Est.Error       Q2.5     Q97.5
## R2 0.06594951 0.01680781 0.03504214 0.1020468

4.3.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

4.3.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.3.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

4.3.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.3.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: productivity ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedinflow                18.47     37.70   -56.05    91.52 1.00     3683     2748
## Ninitial4:StageWithseedinflow                -7.56     25.81   -57.57    43.36 1.00     3347     2581
## Ninitial8:StageWithseedinflow                 3.49     25.38   -46.93    52.91 1.00     3560     2610
## Ninitial16:StageWithseedinflow               58.70     23.40    13.04   106.65 1.00     3809     2744
## Ninitial32:StageWithseedinflow               48.44     30.05   -10.79   105.72 1.00     3673     2738
## Ninitial2:StageWithoutseedinflow             64.80      6.57    51.70    77.79 1.00     3516     2795
## Ninitial4:StageWithoutseedinflow             25.63     11.17     4.53    47.52 1.00     3151     2517
## Ninitial8:StageWithoutseedinflow             37.11     11.78    14.09    60.26 1.00     3568     2570
## Ninitial16:StageWithoutseedinflow            65.50     21.54    23.32   107.83 1.00     3754     2854
## Ninitial32:StageWithoutseedinflow            79.00     33.06    15.36   144.51 1.00     3660     2525
## Ninitial2:StageWithseedinflow:Shannon        18.26     22.59   -25.79    62.96 1.00     3682     2746
## Ninitial4:StageWithseedinflow:Shannon        25.33     11.31     2.89    47.30 1.00     3349     2616
## Ninitial8:StageWithseedinflow:Shannon        17.57      8.97    -0.04    35.41 1.00     3542     2506
## Ninitial16:StageWithseedinflow:Shannon       -1.02      6.94   -15.25    12.66 1.00     3809     2724
## Ninitial32:StageWithseedinflow:Shannon        2.46      7.90   -12.69    17.98 1.00     3655     2823
## Ninitial2:StageWithoutseedinflow:Shannon     -9.94      4.07   -18.06    -1.85 1.00     3515     2707
## Ninitial4:StageWithoutseedinflow:Shannon     11.53      5.22     1.30    21.34 1.00     3167     2553
## Ninitial8:StageWithoutseedinflow:Shannon      6.08      4.49    -2.76    14.85 1.00     3589     2544
## Ninitial16:StageWithoutseedinflow:Shannon    -3.60      6.77   -16.87     9.67 1.00     3732     2860
## Ninitial32:StageWithoutseedinflow:Shannon    -5.79      9.40   -24.41    12.40 1.00     3657     2525
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     6.73      0.19     6.37     7.11 1.00     5563     2621
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.2345123 0.02483178 0.1862827 0.2825331

4.3.2.1 Posterior predictive checks

4.3.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.3.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

4.3.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.4 Forest1

Rüger, N., R. Condit, D. H. Dent, S. J. DeWalt, S. P. Hubbell, J. W. Lichstein, O. R. Lopez, C. Wirth, and C. E. Farrior. 2020. Demographic trade-offs predict tropical forest dynamics. Science 368:165–168.

4.4.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: productivity ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow               31.35      2.03    27.44    35.31 1.00     2048     2039
## StageWithoutseedinflow            27.91      2.25    23.51    32.33 1.00     2161     2262
## StageWithseedinflow:Shannon        3.76      1.21     1.42     6.09 1.00     2131     2002
## StageWithoutseedinflow:Shannon     3.81      1.66     0.63     7.06 1.00     2165     2334
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    13.36      0.34    12.73    14.08 1.00     2845     2437
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##      Estimate  Est.Error       Q2.5      Q97.5
## R2 0.04752129 0.01399466 0.02318427 0.07695434

4.4.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

4.4.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.4.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

4.4.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.4.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: productivity ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedinflow                43.41      7.61    28.58    58.21 1.00     4339     3259
## Ninitial4:StageWithseedinflow                49.09      6.37    36.67    61.51 1.00     4208     2878
## Ninitial8:StageWithseedinflow                47.12      7.41    32.95    61.80 1.00     4085     3033
## Ninitial16:StageWithseedinflow               36.67      9.41    18.63    55.35 1.00     4348     3192
## Ninitial32:StageWithseedinflow               17.89     14.05    -9.71    45.79 1.00     4288     3195
## Ninitial2:StageWithoutseedinflow             45.29      9.53    26.50    64.19 1.00     3924     3113
## Ninitial4:StageWithoutseedinflow             42.32      6.07    30.55    54.00 1.00     3769     2662
## Ninitial8:StageWithoutseedinflow             52.11      6.75    38.58    65.01 1.00     4158     2544
## Ninitial16:StageWithoutseedinflow            18.27      6.43     5.88    31.12 1.00     4692     2921
## Ninitial32:StageWithoutseedinflow            10.35      5.88    -1.24    21.77 1.00     4147     2893
## Ninitial2:StageWithseedinflow:Shannon        -7.99      6.35   -20.51     4.59 1.00     4315     2984
## Ninitial4:StageWithseedinflow:Shannon        -8.62      4.76   -17.96     0.59 1.00     4284     2986
## Ninitial8:StageWithseedinflow:Shannon        -4.51      4.79   -13.84     4.45 1.00     4147     3079
## Ninitial16:StageWithseedinflow:Shannon        1.97      4.75    -7.37    11.08 1.00     4344     2965
## Ninitial32:StageWithseedinflow:Shannon        8.92      5.70    -2.40    20.21 1.00     4329     3154
## Ninitial2:StageWithoutseedinflow:Shannon    -13.79      8.93   -31.71     3.83 1.00     3908     3059
## Ninitial4:StageWithoutseedinflow:Shannon     -7.38      4.94   -16.91     2.27 1.00     3652     2962
## Ninitial8:StageWithoutseedinflow:Shannon    -12.28      5.03   -21.98    -2.17 1.00     4150     2767
## Ninitial16:StageWithoutseedinflow:Shannon    10.99      4.11     2.58    18.86 1.00     4802     2988
## Ninitial32:StageWithoutseedinflow:Shannon    12.64      3.27     6.38    18.97 1.00     3871     2812
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    11.84      0.34    11.22    12.53 1.00     7498     2740
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate Est.Error      Q2.5     Q97.5
## R2 0.1468626 0.0223651 0.1037254 0.1903548

4.4.2.1 Posterior predictive checks

4.4.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.4.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

4.4.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.5 Forest2

Maréchaux, I., and J. Chave. 2017. An individual-based forest model to jointly simulate carbon and tree diversity in Amazonia: description and applications. Ecological Monographs.

4.5.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: productivity ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow               91.66      0.49    90.70    92.62 1.00     1847     1983
## StageWithoutseedinflow            71.71      0.56    70.62    72.83 1.00     1921     1977
## StageWithseedinflow:Shannon        0.55      0.18     0.20     0.91 1.00     1858     1742
## StageWithoutseedinflow:Shannon     0.19      0.30    -0.43     0.78 1.00     1795     2165
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     3.56      0.09     3.38     3.74 1.00     3153     2601
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate   Est.Error      Q2.5     Q97.5
## R2 0.8971751 0.002213982 0.8923178 0.9010684

4.5.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

4.5.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.5.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

4.5.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.5.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: productivity ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedinflow                98.48      3.71    91.40   105.70 1.00     3459     2909
## Ninitial4:StageWithseedinflow                99.78      4.37    91.19   108.39 1.00     3693     2815
## Ninitial8:StageWithseedinflow                94.85      6.35    82.42   107.51 1.00     3396     2718
## Ninitial16:StageWithseedinflow               96.22     10.52    75.07   116.94 1.00     3823     2796
## Ninitial32:StageWithseedinflow               93.11     15.33    62.48   123.41 1.00     3783     2912
## Ninitial2:StageWithoutseedinflow             67.57      2.07    63.48    71.54 1.00     4180     3007
## Ninitial4:StageWithoutseedinflow             68.60      1.96    64.57    72.43 1.00     3329     3059
## Ninitial8:StageWithoutseedinflow             73.93      2.14    69.75    78.02 1.00     3367     2993
## Ninitial16:StageWithoutseedinflow            66.25      2.02    62.23    70.18 1.00     3875     3274
## Ninitial32:StageWithoutseedinflow            71.19      2.69    65.84    76.41 1.00     3790     3083
## Ninitial2:StageWithseedinflow:Shannon        -4.02      2.43    -8.81     0.59 1.00     3420     2828
## Ninitial4:StageWithseedinflow:Shannon        -2.74      2.06    -6.80     1.29 1.00     3673     2754
## Ninitial8:StageWithseedinflow:Shannon        -0.34      2.37    -5.06     4.29 1.00     3399     2709
## Ninitial16:StageWithseedinflow:Shannon       -0.85      3.14    -7.09     5.44 1.00     3821     2947
## Ninitial32:StageWithseedinflow:Shannon        0.02      3.90    -7.66     7.74 1.00     3778     3030
## Ninitial2:StageWithoutseedinflow:Shannon      3.85      1.53     0.88     6.83 1.00     4228     2889
## Ninitial4:StageWithoutseedinflow:Shannon      3.04      1.23     0.67     5.50 1.00     3340     2885
## Ninitial8:StageWithoutseedinflow:Shannon     -0.76      1.07    -2.84     1.32 1.00     3439     2781
## Ninitial16:StageWithoutseedinflow:Shannon     2.68      0.95     0.85     4.59 1.00     3884     3221
## Ninitial32:StageWithoutseedinflow:Shannon    -0.09      1.09    -2.22     2.08 1.00     3786     2964
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     2.97      0.09     2.81     3.15 1.00     7693     2799
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate   Est.Error      Q2.5     Q97.5
## R2 0.9274615 0.001592317 0.9240772 0.9302001

4.5.2.1 Posterior predictive checks

4.5.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.5.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

4.5.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.6 Dryland

Reineking, B., M. Veste, C. Wissel, and A. Huth. 2006. Environmental variability and allocation trade-offs acrosstain species diversity in a process-based model of succulent plant communities. Ecological Modelling.

4.6.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: productivity ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow               63.05      1.51    60.19    66.16 1.00     1829     2059
## StageWithoutseedinflow            43.49      2.03    39.62    47.61 1.01     1809     1841
## StageWithseedinflow:Shannon        2.02      0.64     0.71     3.22 1.00     1858     1850
## StageWithoutseedinflow:Shannon     4.55      1.28     2.01     7.01 1.01     1745     1852
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    10.80      0.29    10.28    11.40 1.00     2810     2178
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.3977007 0.02144508 0.3557756 0.4375859

4.6.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

4.6.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.6.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

4.6.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.6.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: productivity ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedinflow               101.29      6.40    88.85   113.89 1.00     3423     2588
## Ninitial4:StageWithseedinflow                95.54      8.95    78.23   113.33 1.00     4234     2815
## Ninitial8:StageWithseedinflow               105.48     11.14    83.62   127.02 1.00     3656     2891
## Ninitial16:StageWithseedinflow              103.41     15.96    72.09   134.72 1.00     3677     3089
## Ninitial32:StageWithseedinflow              131.68     20.32    92.09   171.10 1.00     3509     3222
## Ninitial2:StageWithoutseedinflow             50.37      4.29    41.93    58.93 1.00     3758     2898
## Ninitial4:StageWithoutseedinflow             66.04      4.53    57.07    75.00 1.00     3849     2941
## Ninitial8:StageWithoutseedinflow             63.88      4.69    54.52    73.23 1.00     4032     3020
## Ninitial16:StageWithoutseedinflow            79.92      6.62    67.00    93.15 1.00     4063     2945
## Ninitial32:StageWithoutseedinflow            77.72      9.58    59.69    97.23 1.00     3830     2898
## Ninitial2:StageWithseedinflow:Shannon       -24.71      4.41   -33.44   -15.98 1.00     3578     2634
## Ninitial4:StageWithseedinflow:Shannon       -14.05      4.69   -23.34    -4.95 1.00     4222     2430
## Ninitial8:StageWithseedinflow:Shannon       -15.35      4.58   -24.21    -6.30 1.00     3628     2871
## Ninitial16:StageWithseedinflow:Shannon      -11.46      5.40   -22.05    -0.91 1.00     3678     3110
## Ninitial32:StageWithseedinflow:Shannon      -17.75      5.87   -29.22    -6.29 1.00     3519     3197
## Ninitial2:StageWithoutseedinflow:Shannon     -3.37      3.37   -10.05     3.20 1.00     3823     2987
## Ninitial4:StageWithoutseedinflow:Shannon    -10.29      3.13   -16.38    -4.10 1.00     3807     2880
## Ninitial8:StageWithoutseedinflow:Shannon     -7.05      2.79   -12.50    -1.60 1.00     4062     3171
## Ninitial16:StageWithoutseedinflow:Shannon   -13.88      3.56   -20.92    -6.83 1.00     4111     3090
## Ninitial32:StageWithoutseedinflow:Shannon   -11.14      4.77   -20.91    -2.10 1.00     3828     2731
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     7.97      0.22     7.56     8.42 1.00     6281     2556
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate Est.Error      Q2.5    Q97.5
## R2 0.5794212 0.0160797 0.5456339 0.608345

4.6.2.1 Posterior predictive checks

4.6.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.6.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

4.6.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.